Abstract
This article discusses translation planes of dimension two and characteristic two. Let G be a subgroup of the linear translation complement of such a plane π. The nature of G and its possible action on π are investigated. This continues previous work of the authors. It is shown that no new groups occur.
Highlights
This article discusses translation planes of dimension two and characteristic two
A translation plane of dimension two over GF(q) F may be represented by a vector space of dimension four over F
A complete classification may not be possible if this is understood to include an enumeration of all existing planes
Summary
We are concerned with the case where the translation complement contains elations. In one of his cases, G has a normal subgroup of index two and odd order. In unpublished work he has shown that, under much broader circumstances than we have here, G must be dihedral. Let O be an involution and T an element of odd order such that has index 2 in . -i the group generated by the involutions is dihedral. If H is faithful on V I and is reducible on V I it has at least two invarlant 1-spaces and is a subgroup of the direct product of two cyclic groups. In Hering’s remaining case, G has a subgroup of odd order and index 2. By Hering [7], Theorem (5.1) in case (b) of (3.4), G has an invariant subplane which is a Lneburg plane
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